Dirichlet series
| L(s) = 1 | + 16·7-s + 8·19-s − 216·25-s + 128·43-s − 304·49-s − 104·61-s − 88·73-s − 1.18e3·121-s + 127-s + 131-s + 128·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.70e3·169-s + 173-s − 3.45e3·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 16/7·7-s + 8/19·19-s − 8.63·25-s + 2.97·43-s − 6.20·49-s − 1.70·61-s − 1.20·73-s − 9.81·121-s + 0.00787·127-s + 0.00763·131-s + 0.962·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 10.0·169-s + 0.00578·173-s − 19.7·175-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{40} \cdot 19^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{40} \cdot 19^{20}\right)^{s/2} \, \Gamma_{\C}(s+1)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{80} \cdot 3^{40} \cdot 19^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.81204\times 10^{37}\) |
| Root analytic conductor: | \(8.63426\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{80} \cdot 3^{40} \cdot 19^{20} ,\ ( \ : [1]^{20} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(1.434142975\times10^{-5}\) |
| \(L(\frac12)\) | \(\approx\) | \(1.434142975\times10^{-5}\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| 19 | \( ( 1 - 4 T - 99 T^{2} + 2080 T^{3} - 626 p T^{4} - 40 p^{3} T^{5} - 626 p^{3} T^{6} + 2080 p^{4} T^{7} - 99 p^{6} T^{8} - 4 p^{8} T^{9} + p^{10} T^{10} )^{2} \) | |
| good | 5 | \( ( 1 + 108 T^{2} + 5678 T^{4} + 41038 p T^{6} + 1170509 p T^{8} + 148729404 T^{10} + 1170509 p^{5} T^{12} + 41038 p^{9} T^{14} + 5678 p^{12} T^{16} + 108 p^{16} T^{18} + p^{20} T^{20} )^{2} \) |
| 7 | \( ( 1 - 4 T + 116 T^{2} - 72 p T^{3} + 7283 T^{4} - 32280 T^{5} + 7283 p^{2} T^{6} - 72 p^{5} T^{7} + 116 p^{6} T^{8} - 4 p^{8} T^{9} + p^{10} T^{10} )^{4} \) | |
| 11 | \( ( 1 + 54 p T^{2} + 203762 T^{4} + 4355420 p T^{6} + 69959293 p^{2} T^{8} + 1156874862468 T^{10} + 69959293 p^{6} T^{12} + 4355420 p^{9} T^{14} + 203762 p^{12} T^{16} + 54 p^{17} T^{18} + p^{20} T^{20} )^{2} \) | |
| 13 | \( ( 1 - 850 T^{2} + 31761 p T^{4} - 136650136 T^{6} + 33923074226 T^{8} - 6479948417196 T^{10} + 33923074226 p^{4} T^{12} - 136650136 p^{8} T^{14} + 31761 p^{13} T^{16} - 850 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 17 | \( ( 1 + 1644 T^{2} + 1390958 T^{4} + 781518966 T^{6} + 325424327329 T^{8} + 105598372288348 T^{10} + 325424327329 p^{4} T^{12} + 781518966 p^{8} T^{14} + 1390958 p^{12} T^{16} + 1644 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 23 | \( ( 1 + 3352 T^{2} + 5370973 T^{4} + 5512416384 T^{6} + 4129771232834 T^{8} + 2431928446462928 T^{10} + 4129771232834 p^{4} T^{12} + 5512416384 p^{8} T^{14} + 5370973 p^{12} T^{16} + 3352 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 29 | \( ( 1 - 2448 T^{2} + 3174333 T^{4} - 3611948032 T^{6} + 3771587751314 T^{8} - 3368366451496928 T^{10} + 3771587751314 p^{4} T^{12} - 3611948032 p^{8} T^{14} + 3174333 p^{12} T^{16} - 2448 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 31 | \( ( 1 - 5606 T^{2} + 15046317 T^{4} - 26204827848 T^{6} + 34130982371538 T^{8} - 35974435861206692 T^{10} + 34130982371538 p^{4} T^{12} - 26204827848 p^{8} T^{14} + 15046317 p^{12} T^{16} - 5606 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 37 | \( ( 1 - 9130 T^{2} + 40328189 T^{4} - 115007922232 T^{6} + 236614078694130 T^{8} - 369183565637344188 T^{10} + 236614078694130 p^{4} T^{12} - 115007922232 p^{8} T^{14} + 40328189 p^{12} T^{16} - 9130 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 41 | \( ( 1 - 56 p T^{2} + 7480061 T^{4} - 16740897536 T^{6} + 35839012442018 T^{8} - 55287780174499088 T^{10} + 35839012442018 p^{4} T^{12} - 16740897536 p^{8} T^{14} + 7480061 p^{12} T^{16} - 56 p^{17} T^{18} + p^{20} T^{20} )^{2} \) | |
| 43 | \( ( 1 - 32 T + 4044 T^{2} - 137316 T^{3} + 10369947 T^{4} - 392349512 T^{5} + 10369947 p^{2} T^{6} - 137316 p^{4} T^{7} + 4044 p^{6} T^{8} - 32 p^{8} T^{9} + p^{10} T^{10} )^{4} \) | |
| 47 | \( ( 1 + 15442 T^{2} + 117782178 T^{4} + 576878603060 T^{6} + 1999767554065989 T^{8} + 5112919411545667204 T^{10} + 1999767554065989 p^{4} T^{12} + 576878603060 p^{8} T^{14} + 117782178 p^{12} T^{16} + 15442 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 53 | \( ( 1 - 5272 T^{2} + 20157069 T^{4} - 80086810624 T^{6} + 287322232118306 T^{8} - 864909000275559888 T^{10} + 287322232118306 p^{4} T^{12} - 80086810624 p^{8} T^{14} + 20157069 p^{12} T^{16} - 5272 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 59 | \( ( 1 - 14706 T^{2} + 103595677 T^{4} - 477611439960 T^{6} + 1673405319446226 T^{8} - 5513338543279463404 T^{10} + 1673405319446226 p^{4} T^{12} - 477611439960 p^{8} T^{14} + 103595677 p^{12} T^{16} - 14706 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 61 | \( ( 1 + 26 T + 14858 T^{2} + 365004 T^{3} + 98333501 T^{4} + 1998734788 T^{5} + 98333501 p^{2} T^{6} + 365004 p^{4} T^{7} + 14858 p^{6} T^{8} + 26 p^{8} T^{9} + p^{10} T^{10} )^{4} \) | |
| 67 | \( ( 1 - 106 p T^{2} + 44670525 T^{4} - 190943988456 T^{6} + 860811526643730 T^{8} - 4404565062380249140 T^{10} + 860811526643730 p^{4} T^{12} - 190943988456 p^{8} T^{14} + 44670525 p^{12} T^{16} - 106 p^{17} T^{18} + p^{20} T^{20} )^{2} \) | |
| 71 | \( ( 1 - 32514 T^{2} + 542192333 T^{4} - 5903648857240 T^{6} + 46075143152194194 T^{8} - \)\(26\!\cdots\!16\)\( T^{10} + 46075143152194194 p^{4} T^{12} - 5903648857240 p^{8} T^{14} + 542192333 p^{12} T^{16} - 32514 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 73 | \( ( 1 + 22 T + 20522 T^{2} + 220572 T^{3} + 184934141 T^{4} + 1101199756 T^{5} + 184934141 p^{2} T^{6} + 220572 p^{4} T^{7} + 20522 p^{6} T^{8} + 22 p^{8} T^{9} + p^{10} T^{10} )^{4} \) | |
| 79 | \( ( 1 - 18706 T^{2} + 178339661 T^{4} - 1063664021272 T^{6} + 3987225811554930 T^{8} - 14272091785021581228 T^{10} + 3987225811554930 p^{4} T^{12} - 1063664021272 p^{8} T^{14} + 178339661 p^{12} T^{16} - 18706 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 83 | \( ( 1 + 12664 T^{2} + 114901229 T^{4} + 832893595776 T^{6} + 6296966398094242 T^{8} + 43117565147459530512 T^{10} + 6296966398094242 p^{4} T^{12} + 832893595776 p^{8} T^{14} + 114901229 p^{12} T^{16} + 12664 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 89 | \( ( 1 - 44024 T^{2} + 998391357 T^{4} - 15348835664896 T^{6} + 176380996880684322 T^{8} - \)\(15\!\cdots\!88\)\( T^{10} + 176380996880684322 p^{4} T^{12} - 15348835664896 p^{8} T^{14} + 998391357 p^{12} T^{16} - 44024 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
| 97 | \( ( 1 - 41426 T^{2} + 958003437 T^{4} - 16332775911512 T^{6} + 214384906635230098 T^{8} - \)\(22\!\cdots\!84\)\( T^{10} + 214384906635230098 p^{4} T^{12} - 16332775911512 p^{8} T^{14} + 958003437 p^{12} T^{16} - 41426 p^{16} T^{18} + p^{20} T^{20} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−1.68193614713807994801197847575, −1.67201791354769404312312019091, −1.54284707481672771465989172032, −1.49247768136003596822311782386, −1.40105046443239875838477532808, −1.38942788513993473551935742028, −1.25107358308354890092901040471, −1.22169233767170067078461816722, −1.15845264224576012330772915674, −1.10385470270614744204136383883, −1.09686150668029726456134165598, −0.931980881400707529899111194723, −0.854879300747292382723430158308, −0.814114202080014157251476945571, −0.792122129378495490042380955822, −0.68071335451519600193094066524, −0.67604616745932114795265115783, −0.48284087719310112163941601555, −0.39391122653925277636771213659, −0.26615013656272162272526789851, −0.23564365144164671560746182245, −0.22305541379725383830432910080, −0.14098781794414136270204552200, −0.00553327715345086477110533393, −0.00528375849895951542518361141, 0.00528375849895951542518361141, 0.00553327715345086477110533393, 0.14098781794414136270204552200, 0.22305541379725383830432910080, 0.23564365144164671560746182245, 0.26615013656272162272526789851, 0.39391122653925277636771213659, 0.48284087719310112163941601555, 0.67604616745932114795265115783, 0.68071335451519600193094066524, 0.792122129378495490042380955822, 0.814114202080014157251476945571, 0.854879300747292382723430158308, 0.931980881400707529899111194723, 1.09686150668029726456134165598, 1.10385470270614744204136383883, 1.15845264224576012330772915674, 1.22169233767170067078461816722, 1.25107358308354890092901040471, 1.38942788513993473551935742028, 1.40105046443239875838477532808, 1.49247768136003596822311782386, 1.54284707481672771465989172032, 1.67201791354769404312312019091, 1.68193614713807994801197847575
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.